Talk:Fibre homotopy trivial bundles (Ex)
We consider 5-dimensional real vector bundles over . Isomorphism classes of these are given by their clutching function in Given that is isomorphic to the surjection , we see that the vector bundle corresponding to times the generator has a sphere bundle which is fiber homotopically trivial, so in particular we have homotopy equivalences . From another exercise we know that stably . From a third exercise we know that the first Pontryagin class of is . It follows that the first Pontryagin class of is non-trivial, since under the map just corresponds to projection to one factor. Hence is a homotopy equivalence which doesn't preserve the first Pontryagin class, as has stably trivial tangent bundle, hence trivial .
Similarly one can argue with -dimensional vector bundles over ; the J-homomorphism has always a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces homotopy equivalences which do not preserve .